Additional Symmetries of Invariant Complex Domains

Definitions and mathematical statement

Let $K$ be a connected compact Lie group, let $L\leq K$ be closed, and put \[ X=K/L,\qquad X_{{\mathbb{C}}}=K_{{\mathbb{C}}}/L_{{\mathbb{C}}}, \] with the intended complexifications. Let $D\subset X_{{\mathbb{C}}}$ be a nonempty connected open set invariant under $K$. Let $\operatorname{Aut}(D)$ be the group of biholomorphic automorphisms, with its intended compact-open group topology, and let \[ \rho:K\longrightarrow\operatorname{Aut}(D) \] be the induced action. The evident group is its image $\rho(K)\cong K/\ker\rho$; faithfulness is not stated. Since $K$ is connected, this image lies in the identity component $\operatorname{Aut}(D)^0$.

The source question is whether there can be additional positive-dimensional automorphism symmetries beyond $\rho(K)$. Under hypotheses making $\operatorname{Aut}(D)$ a finite-dimensional Lie group, a precise identity-component version asks whether \[ \rho(K)\subsetneq\operatorname{Aut}(D)^0, \] can occur, equivalently whether its real Lie algebra has directions beyond those induced by $K$. Extra disconnected automorphisms alone do not constitute this identity-component phenomenon. The requested task may be viewed as deciding this for the stated invariant domains and describing when it occurs; the source gives no sharper classification or universal exclusion claim.

The record does not impose relative compactness, boundary regularity, hyperbolicity, effectiveness or other hypotheses on $D$, nor does it supply assumptions ensuring a finite-dimensional automorphism Lie group. Without that framework, the meaning of positive-dimensional symmetry must be specified, for example in terms of connected finite-dimensional transformation subgroups. The choices and quotient properties of the complexifications also need confirmation. No invariance of $D$ under all of $K_{{\mathbb{C}}}$ is assumed merely from $K$-invariance.

The supplied assessment checked 2026-08-21 labels the question open and says compactness results for automorphism groups do not settle this general identity-component question. Compactness of an automorphism group would not by itself identify its connected subgroup $\rho(K)$ with the whole identity component. This is the source's stated limitation, not an asserted new theorem about these domains.

Short English statement

Ask whether a $K$-invariant domain in the complexification of $K/L$ has connected automorphism symmetries beyond the effective image of $K$. The intended domain hypotheses and finite-dimensional automorphism-group framework are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID 30001022, source code catalogue entry; supplied complete source record and its corrected-verified notation-cleanup note. Literature checked 2026-08-21: the assessment is supplied dated curation, not independently verified here.
• Komplexe Analysis (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/38, pp. 2165--2218: https://doi.org/10.4171/owr/2008/38. Proposed-year metadata gives 2008.